Finite Group Databasebeta

← Dic3×#4096 #14416268

Subgroups — Dic3×#4096 #14416268

Order 1920.

1000 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×Dic3×#41C211yesmaximal, characteristic, complemented
2C2×C3×#4096C211yesmaximal, characteristic
3Dic3×#709C211yesmaximal, characteristic, complemented
4#11751334C211yesmaximal, characteristic, complemented
5Dic3×#689C212yesmaximal, complemented
6Dic3×#688C212yesmaximal, complemented
7Dic3×#687C212yesmaximal, complemented
8Dic3×#686C212yesmaximal, complemented
9#11748458C211yesmaximal, characteristic, complemented
10#11748364C211yesmaximal, characteristic, complemented
11C2×Dic3×#168C211yesmaximal, characteristic, complemented
12C4×#4096—31nomaximal, Hall
13C2×C3×C5×#41C2211yescharacteristic
14C2×C3×#709C2211yescharacteristic
15#91815C2211yescharacteristic
16C22×C3⋊#168C2211yescharacteristic
17C5×Q8×Dic3C2212yes—
18C5×D4×Dic3—22no—
19C5×D4×Dic3C2212yes—
20C5×#926—21no—
21C5×#926C2211yescharacteristic
22C2×C4×C5×Dic3—21no—
23C2×C4×C5×Dic3C2211yescharacteristic
24C3×#4096C418yescomplemented
25C2×C3×#689C2211yescharacteristic
26C2×C3×#688C2211yescharacteristic
27C2×C3×#687C2211yescharacteristic
28C2×C3×#686C2211yescharacteristic
29C22×C3×#168C2211yescharacteristic
30Dic3×C22⋊D5—22no—
31Dic3×D20C2212yes—
32C4×D5×Dic3—21no—
33#91357—21no—
34Dic3×Dic10C2212yes—
35#91119C2212yes—
36#91116C2212yes—
37#90988C2212yes—
38#90985C2212yes—
39#90970C2212yes—
40#90964C2212yes—
41Dic3×#168C2212yes—
42Dic3×#168C2212yes—
43Dic3×#135—51nomaximal, Hall
44C4×C5×#41—31no—
45C2×#4096D311yescharacteristic
46C4×#709—31no—
47#65522—31no—
48C4×#689—32no—
49C4×#688—32no—
50C4×#687—32no—
51C4×#686—32no—
52#65291—31no—
53#65287—31no—
54C2×C4×#168—31no—
55C3×C5×#41—22no—
56C3×C5×#41C2×C414yescomplemented
57C2×C3×C5×Q8C2311yescharacteristic
58C2×C3×C5×D4—21no—
59C2×C3×C5×D4C2311yescharacteristic
60C22×C3×C4×C5—21no—
61C22×C3×C4×C5C2311yescharacteristic
62C22×C5×Dic3—22no—
63C22×C5×Dic3D412yes—
64C2×C3×C22⋊D5—21no—
65C3×#709—22no—
66C3×#709C2×C414yescomplemented
67C2×C3×D20C2311yescharacteristic
68C2×C3×C4×D5—21no—
69C2×C3×Dic10C2311yescharacteristic
70C2×D5×Dic3—22no—
71#6719—21no—
72C4⋊Dic15C2312yes—
73C4×Dic15—21no—
74C2×C3⋊#168—22no—
75C2×C3⋊#168C2312yes—
76C5×#217—21no—
77C5×#217—22no—
78C5×C4⋊Dic3—22no—
79C5×C4⋊Dic3C2312yes—
80C4×C5×Dic3—21no—
81C4×C5×Dic3—22no—
82C4×C5×Dic3D411yescharacteristic
83C4×C5×Dic3D411yescharacteristic
84C4×C5×Dic3C2312yes—
85C3×#689C2×C414yescomplemented
86C3×#688C2×C414yescomplemented
87C3×#687C2×C414yescomplemented
88C3×#686C2×C414yescomplemented
89C2×C3×#168C2×C414yescomplemented
90C2×C3×#168C2311yescharacteristic
91C2×C3×#168C2311yescharacteristic
92#6668—22no—
93#6665—22no—
94Dic3×Dic5—22no—
95C2×C3×#135—51no—
96Dic3×#41—51no—
97Dic3×#41D511yescharacteristic, complemented
98#5233—51no—
99#5233—51no—
100Dic3×Q16—52no—
101Dic3×SD16—52no—
102Dic3×SD16—52no—
103Dic3×D8—52no—
104#5106—51no—
105C2×C8×Dic3—51no—
106C2×C5×#41C2×D311yescharacteristic
107C2×#709C2×D311yescharacteristic
108C4×C5×Q8—32no—
109C4×C5×D4—32no—
110C4×C5×D4—62no—
111C5×#117—31no—
112C5×#117—61no—
113C2×C42×C5—31no—
114C2×C42×C5—61no—
115#4096Dic318yesdirect product factor
116C2×#689C2×D311yescharacteristic
117C2×#688C2×D311yescharacteristic
118C2×#687C2×D311yescharacteristic
119C2×#686C2×D311yescharacteristic
120C4×C22⋊D5—62no—
121#4072—31no—
122C22×#168—31no—
123C22×#168C2×D311yescharacteristic
124C4×D20—32no—
125#4018—61no—
126C42×D5—61no—
127C4×Dic10—32no—
128#3967—32no—
129#3964—32no—
130#3942—32no—
131#3941—32no—
132#3940—32no—
133#3939—32no—
134C4×#168—32no—
135C4×#168—32no—
136C4×#135—151noSylow
137C23×C3×C5—21no—
138C23×C3×C5C2×D411yescharacteristic
139C22×C3×D5—21no—
140C3×C5×Q8—21no—
141C3×C5×Q8C22×C412yes—
142C3×C5×D4—24no—
143C3×C5×D4—21no—
144C3×C5×D4C22×C412yes—
145C2×C3×C4×C5—21no—
146C2×C3×C4×C5—24no—
147C2×C3×C4×C5C22×C412yes—
148C2×C3×C4×C5C2×D411yescharacteristic
149C2×C3×C4×C5#4112yes—
150C2×C3×C4×C5C2411yescharacteristic
151C2×Dic15—22no—
152C2×C5×Dic3—24no—
153C2×C5×Dic3—22no—
154C2×C5×Dic3—44no—
155C2×C5×Dic3C2×D412yes—
156C2×C5×Dic3C2×D412yes—
157C3×C22⋊D5—24no—
158C2×C3×Dic5—21no—
159C3×D20—21no—
160C3×D20C22×C412yes—
161C3×C4×D5—24no—
162C3×Dic10—21no—
163C3×Dic10C22×C412yes—
164D5×Dic3—44no—
165C3⋊#168—22no—
166C3×#168C22×C412yes—
167C3×#168C22×C412yes—
168C2×C3×#41—51no—
169C2×C3×#41C2×D511yescharacteristic
170C3×#135—58no—
171C2×C3×Q16—51no—
172C2×C3×SD16—51no—
173C2×C3×SD16—51no—
174C2×C3×D8—51no—
175C22×C3×C8—51no—
176Q8×Dic3—52no—
177Q8×Dic3C2×D512yes—
178D4×Dic3—22no—
179D4×Dic3—52no—
180D4×Dic3—102no—
181D4×Dic3C2×D512yes—
182#926—21no—
183#926—51no—
184#926—101no—
185#926C2×D511yescharacteristic
186C2×C4×Dic3—21no—
187C2×C4×Dic3—101no—
188C2×C4×Dic3C2×D511yescharacteristic
189C22×C3⋊C8—51no—
190#835—52no—
191#835—52no—
192#832—52no—
193#832—52no—
194#818—52no—
195#817—52no—
196C8×Dic3—52no—
197C8×Dic3—52no—
198C5×#41—22no—
199C5×#41C2×Dic314yescomplemented
200C2×C5×Q8C22×D311yescharacteristic
201C2×C5×D4—21no—
202C2×C5×D4C22×D311yescharacteristic
203C22×C4×C5—21no—
204C22×C4×C5—32no—
205C22×C4×C5—62no—
206C22×C4×C5C22×D311yescharacteristic
207C2×C22⋊D5—21no—
208#709—22no—
209#709C2×Dic314yescomplemented
210C2×D20C22×D311yescharacteristic
211C2×C4×D5—21no—
212C2×C4×D5—62no—
213C2×Dic10C22×D311yescharacteristic
214C5×C4⋊C4—32no—
215C5×C4⋊C4—62no—
216C5×C22⋊C4—62no—
217C5×C22⋊C4—61no—
218C42×C5—31no—
219C42×C5—31no—
220C42×C5—32no—
221C42×C5—62no—
222C42×C5—61no—
223C22⋊Dic5—61no—
224#689C2×Dic314yescomplemented
225#688C2×Dic314yescomplemented
226#687C2×Dic314yescomplemented
227#686C2×Dic314yescomplemented
228#685—62no—
229C4⋊Dic5—32no—
230#683—62no—
231C4×Dic5—62no—
232C4×Dic5—61no—
233C2×#168—32no—
234C2×#168—62no—
235C2×#168C2×Dic314yescomplemented
236C2×#168C22×D311yescharacteristic
237C2×#168C22×D311yescharacteristic
238C2×#135—51no—
239C4×#41—31no—
240C4×#41—151no—
241C4×Q16—152no—
242C4×SD16—152no—
243C4×SD16—152no—
244C4×D8—152no—
245#427—151no—
246#416—151no—
247#416—151no—
248C2×C4×C8—151no—
249C22×C3×C5—22no—
250C22×C3×C5—22no—
251C22×C3×C5—42no—
252C22×C3×C5C4×D412yes—
253C22×C3×C5C22×D411yescharacteristic
254C2×C3×D5—22no—
255C2×C3×D5—42no—
256C3×C4×C5—22no—
257C3×C4×C5C4×D412yes—
258C3×C4×C5C23×C411yescharacteristic
259C3×C4×C5C2×#4111yescharacteristic
260Dic15—42no—
261C3×Dic5—22no—
262C5×Dic3—22no—
263C5×Dic3—42no—
264C5×Dic3#13514yescomplemented
265C3×#41—22no—
266C3×#41—54no—
267C3×#41—102no—
268C3×#41C4×D514yescomplemented
269C2×C3×Q8—51no—
270C2×C3×Q8C22×D511yescharacteristic
271C2×C3×D4—21no—
272C2×C3×D4—51no—
273C2×C3×D4—101no—
274C2×C3×D4C22×D511yescharacteristic
275C22×C3×C4—21no—
276C22×C3×C4—101no—
277C22×C3×C4C22×D511yescharacteristic
278C22×Dic3—22no—
279C22×Dic3—102no—
280C22×Dic3C22⋊D512yes—
281C3×Q16—54no—
282C3×SD16—54no—
283C3×SD16—54no—
284C3×D8—54no—
285C2×C3×C8—54no—
286C2×C3×C8—51no—
287C2×C3×C8—51no—
288#217—21no—
289#217—22no—
290#217—101no—
291#217—102no—
292C4⋊Dic3—22no—
293C4⋊Dic3—52no—
294C4⋊Dic3—102no—
295C4⋊Dic3C22×D512yes—
296C4×Dic3—21no—
297C4×Dic3—22no—
298C4×Dic3—101no—
299C4×Dic3—102no—
300C4×Dic3C22⋊D511yescharacteristic
301C4×Dic3C22⋊D511yescharacteristic
302C4×Dic3C22×D512yes—
303C2×C3⋊C8—52no—
304C2×C3⋊C8—102no—
305C23×C5—21no—
306C23×C5D3×D411yescharacteristic
307C22×D5—21no—
308C5×Q8—21no—
309C5×Q8C22×Dic312yes—
310C5×D4—24no—
311C5×D4—21no—
312C5×D4C22×Dic312yes—
313C2×C4×C5—21no—
314C2×C4×C5—24no—
315C2×C4×C5—32no—
316C2×C4×C5—32no—
317C2×C4×C5—64no—
318C2×C4×C5—62no—
319C2×C4×C5—124no—
320C2×C4×C5D3×D411yescharacteristic
321C2×C4×C5#23712yes—
322C2×C4×C5C22×Dic312yes—
323C2×C4×C5C23×D311yescharacteristic
324C22⋊D5—24no—
325C2×Dic5—21no—
326C2×Dic5—62no—
327D20—21no—
328D20C22×Dic312yes—
329C4×D5—24no—
330C4×D5—124no—
331Dic10—21no—
332Dic10C22×Dic312yes—
333#168—62no—
334#168C22×Dic312yes—
335#168C22×Dic312yes—
336C2×#41—51no—
337C2×#41D3×D511yescharacteristic
338#135—58no—
339C2×Q16—51no—
340C2×SD16—51no—
341C2×SD16—51no—
342C2×D8—51no—
343C22×C8—51no—
344C22×C8—151no—
345C4×Q8—32no—
346C4×Q8—152no—
347C4×D4—32no—
348C4×D4—62no—
349C4×D4—152no—
350C4×D4—302no—
351#117—31no—
352#117—61no—
353#117—151no—
354#117—301no—
355C2×C42—31no—
356C2×C42—61no—
357C2×C42—301no—
358#107—152no—
359#106—152no—
360#103—152no—
361#103—152no—
362#102—152no—
363#102—152no—
364C4×C8—152no—
365C4×C8—152no—
366C2×C3×C5—22no—
367C2×C3×C5—42no—
368C2×C3×C5C2×C4×D411yescharacteristic
369C2×C3×C5C2×#13512yes—
370C3×D5—42no—
371C23×C3—21no—
372C23×C3—101no—
373C23×C3C2×C22⋊D511yescharacteristic
374C3×Q8—21no—
375C3×Q8—52no—
376C3×Q8—101no—
377C3×Q8C2×C4×D512yes—
378C3×D4—24no—
379C3×D4—21no—
380C3×D4—52no—
381C3×D4—104no—
382C3×D4—101no—
383C3×D4C2×C4×D512yes—
384C2×C3×C4—21no—
385C2×C3×C4—24no—
386C2×C3×C4—101no—
387C2×C3×C4—104no—
388C2×C3×C4C2×C4×D512yes—
389C2×C3×C4#70912yes—
390C2×C3×C4C2×C22⋊D511yescharacteristic
391C2×C3×C4C23×D511yescharacteristic
392C2×Dic3—24no—
393C2×Dic3—22no—
394C2×Dic3—44no—
395C2×Dic3—102no—
396C2×Dic3—204no—
397C2×Dic3C2×C22⋊D512yes—
398C2×Dic3C2×C22⋊D512yes—
399C3×C8—52no—
400C3×C8—52no—
401C3⋊C8—102no—
402C22×C5—22no—
403C22×C5—22no—
404C22×C5—42no—
405C22×C5D4×Dic312yes—
406C22×C5C2×D3×D411yescharacteristic
407C2×D5—22no—
408C2×D5—42no—
409C4×C5—22no—
410C4×C5—34no—
411C4×C5—62no—
412C4×C5—122no—
413C4×C5D4×Dic312yes—
414C4×C5C2×#23711yescharacteristic
415C4×C5C23×Dic311yescharacteristic
416Dic5—22no—
417Dic5—122no—
418#41—22no—
419#41—54no—
420#41—102no—
421#41D5×Dic314yescomplemented
422C2×Q8—51no—
423C2×Q8C2×D3×D511yescharacteristic
424C2×D4—21no—
425C2×D4—51no—
426C2×D4—101no—
427C2×D4C2×D3×D511yescharacteristic
428C22×C4—21no—
429C22×C4—32no—
430C22×C4—62no—
431C22×C4—101no—
432C22×C4—302no—
433C22×C4C2×D3×D511yescharacteristic
434Q16—54no—
435SD16—54no—
436SD16—54no—
437D8—54no—
438C2×C8—54no—
439C2×C8—51no—
440C2×C8—51no—
441C2×C8—152no—
442C2×C8—302no—
443C4⋊C4—32no—
444C4⋊C4—62no—
445C4⋊C4—152no—
446C4⋊C4—302no—
447C22⋊C4—62no—
448C22⋊C4—61no—
449C22⋊C4—302no—
450C22⋊C4—301no—
451C42—31no—
452C42—31no—
453C42—32no—
454C42—62no—
455C42—61no—
456C42—302no—
457C42—301no—
458C3×C5C4×#13511yescharacteristic, complemented, Hall
459C22×C3—22no—
460C22×C3—22no—
461C22×C3—42no—
462C22×C3—102no—
463C22×C3—202no—
464C22×C3C4×C22⋊D512yes—
465C22×C3C22×C22⋊D511yescharacteristic
466C3×C4—22no—
467C3×C4—102no—
468C3×C4C4×C22⋊D512yes—
469C3×C4C22×C4×D511yescharacteristic
470C3×C4C2×#70911yescharacteristic
471Dic3—22no—
472Dic3—42no—
473Dic3—202no—
474Dic3#409614yesdirect product factor
475C2×C5—22no—
476C2×C5—42no—
477C2×C5D3×#13512yes—
478C2×C5C2×D4×Dic311yescharacteristic
479D5—42no—
480C23—21no—
481C23—101no—
482C23D3×C22⋊D511yescharacteristic
483Q8—21no—
484Q8—52no—
485Q8—101no—
486Q8C2×D5×Dic312yes—
487D4—24no—
488D4—21no—
489D4—52no—
490D4—104no—
491D4—101no—
492D4C2×D5×Dic312yes—
493C2×C4—21no—
494C2×C4—24no—
495C2×C4—32no—
496C2×C4—32no—
497C2×C4—64no—
498C2×C4—62no—
499C2×C4—101no—
500C2×C4—104no—
501C2×C4—124no—
502C2×C4—302no—
503C2×C4—604no—
504C2×C4C2×D5×Dic312yes—
505C2×C4#677912yes—
506C2×C4D3×C22⋊D511yescharacteristic
507C2×C4C22×D3×D511yescharacteristic
508C8—52no—
509C8—52no—
510C8—302no—
511C2×C3—22no—
512C2×C3—42no—
513C2×C3—202no—
514C2×C3C2×C4×C22⋊D511yescharacteristic
515C2×C3C2×#409612yes—
516C5Dic3×#13511yescharacteristic, complemented, Sylow
517C22—22no—
518C22—22no—
519C22—42no—
520C22—102no—
521C22—202no—
522C22Dic3×C22⋊D512yes—
523C22C2×D3×C22⋊D511yescharacteristic
524C4—22no—
525C4—34no—
526C4—62no—
527C4—102no—
528C4—122no—
529C4—602no—
530C4Dic3×C22⋊D512yes—
531C4C22×D5×Dic311yescharacteristic
532C4C2×#677911yescharacteristic
533C3C4×#409611yescharacteristic, complemented, Sylow
534C2—22no—
535C2—42no—
536C2—202no—
537C2D3×#409612yes—
538C2C2×Dic3×C22⋊D511yescharacteristic
subgroupG/H
centerC2×C4D3×C22⋊D5
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×#41C22
socleC22×C3×C5C22×D4
prime radicalC2×C3×#4096C2
cosocleC2×C3×C4×C5C24
hypercenterC2×#41D3×D5
hypocenterC3×C5C4×#135
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#41

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C5×Dic3×#41C2semidirect
3. Dic3×#709C2semidirect
4. #11751334C2semidirect
5. Dic3×#689C2semidirect
6. Dic3×#688C2semidirect
7. Dic3×#687C2semidirect
8. Dic3×#686C2semidirect
9. #11748458C2semidirect
10. #11748364C2semidirect
11. C2×Dic3×#168C2semidirect
24. C3×#4096C4semidirect
56. C3×C5×#41C2×C4semidirect
66. C3×#709C2×C4semidirect
85. C3×#689C2×C4semidirect
86. C3×#688C2×C4semidirect
87. C3×#687C2×C4semidirect
88. C3×#686C2×C4semidirect
89. C2×C3×#168C2×C4semidirect
97. Dic3×#41D5semidirect
115. #4096Dic3direct (N × H)
199. C5×#41C2×Dic3semidirect
209. #709C2×Dic3semidirect
224. #689C2×Dic3semidirect
225. #688C2×Dic3semidirect
226. #687C2×Dic3semidirect
227. #686C2×Dic3semidirect
235. C2×#168C2×Dic3semidirect
264. C5×Dic3#135semidirect
268. C3×#41C4×D5semidirect
421. #41D5×Dic3semidirect
458. C3×C5C4×#135semidirect
474. Dic3#4096direct (N × H)
516. C5Dic3×#135semidirect
533. C3C4×#4096semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C5×Dic3×#412
C2×C3×#40962
Dic3×#7092
#117513342
Dic3×#6892
Dic3×#6882
Dic3×#6872
Dic3×#6862
#117484582
#117483642
C2×Dic3×#1682
C4×#40963
Dic3×#1355